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Discrete o-minimal structures

Anand Pillay, Charles Steinhorn
1987 Annals of Pure and Applied Logic  
We assume that the reader is familiar 1 It will be shown in a forthcoming paper by Pillay and Steinhorm that any o-minimal structure has a strongly o-minimal theory.  ...  if every model of T is o-minimal.  ...  of n-tuples in discretely ordered structures with strongly o-minimal theories.  ... 
doi:10.1016/0168-0072(87)90004-2 fatcat:rfoqwoxld5grhfoth2juynuyhi

Compression Schemes, Stable Definable Families, and o-Minimal Structures

H. R. Johnson, M. C. Laskowski
2009 Discrete & Computational Geometry  
As a consequence, the combinatorial complexity (or density) of any * hujohnson@jjay.cuny.edu † Partially supported by NSF grant DMS-0600217. definable family in a structure with a o-minimal theory is bounded  ...  We show that any family of sets uniformly definable in an ominimal structure has an extended compression scheme of size equal to the number of parameters in the defining formula.  ...  Lemma 4. 7 . 7 For any o-minimal structure M and for any formula θ(x; y,z) Theorem 4. 9 . 9 Let M be any o-minimal structure.  ... 
doi:10.1007/s00454-009-9201-3 fatcat:44jjhnaajren5hxl7ijffrieie

Locally o-minimal structures with tame topological properties [article]

Masato Fujita
2021 arXiv   pre-print
We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian  ...  We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional.  ...  Acknowledgment In the initial draft, the author considers structures simultaneously satisfying the property (a) in Definition 1.1 and the properties (b) through (d) in Definition 2.6, ignoring their dependence  ... 
arXiv:2006.13481v3 fatcat:wghfya4plvhyhghzopgthqjxum

O-MINIMALISM

HANS SCHOUTENS
2014 Journal of Symbolic Logic (JSL)  
This paper is devoted to o-minimalism, the study of the first-order properties of o-minimal structures.  ...  To any pseudo-o-minimal structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a nontrivial invariant.  ...  Pseudo-o-minimal structures §10. O-minimalism.  ... 
doi:10.1017/jsl.2013.14 fatcat:hcrqbgyihfej3kt424wfrgeyuu

Locally o-minimal structures and structures with locally o-minimal open core

Antongiulio Fornasiero
2013 Annals of Pure and Applied Logic  
We give a characterisation of structures with locally o-minimal open core, and we show that dense elementary pairs of locally o-minimal structures have locally o-minimal open core.  ...  We study first-order expansions of ordered fields that are definably complete, and moreover either are locally o-minimal, or have a locally o-minimal open core.  ...  One advantage of locally o-minimal structures over o-minimal ones is that they form an elementary class: that is, an ultraproduct of locally o-minimal structures is locally o-minimal.  ... 
doi:10.1016/j.apal.2012.10.002 fatcat:fmwztdzvvva35prhg3qunus77i

The Non-Axiomatizability of O-Minimality [article]

Alex Rennet
2012 arXiv   pre-print
Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures.  ...  This answers negatively a question raised by Schoutens, and shows that the theory consisting of those L-sentences true in all o-minimal L-structures, called the theory of o-minimality (for L), is not recursively  ...  Thus, in the bare field language, pseudoo-minimal structures are o-minimal.  ... 
arXiv:1203.2715v2 fatcat:xjjhehrksndedflcxecq4nrqii

A Penalty Approach for Nonlinear Optimization with Discrete Design Variables [chapter]

Dong K. Shin, Z. Gurdal, O. H. Griffin
1989 Discretization Methods and Structural Optimization — Procedures and Applications  
Although numerous algorithms for structural optimization problems have been developed, relatively little effort has been made for optimum structural design incorporating design variables with discrete  ...  This means only t h e o r i g i n a l o b j e c t i v e function and c o n s t r a i n t penalty terms t a k e p a r t i n t h e minimization process.  ... 
doi:10.1007/978-3-642-83707-4_42 fatcat:hltnvbqhjvhmlaqdzzpgw67rju

O-minimalism [article]

Hans Schoutens
2011 arXiv   pre-print
An ordered structure is called o-minimalistic if it has all the first-order features of an o-minimal structure.  ...  To any o-minimalistic structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a non-trivial invariant.  ...  An o-minimalistic structure with o-minimal open core is itself o-minimal. Proof.  ... 
arXiv:1106.1196v1 fatcat:ik5lb72e6nhgtm3r746en3m7nq

Expansions of o-minimal structures by fast sequences

Harvey Friedman, Chris Miller
2005 Journal of Symbolic Logic (JSL)  
Let ℝ be an o-minimal expansion of (ℝ, <, +) and (ϕ k ) kЄℕ be a sequence of positive real numbers such that limt k →+∞ f (ϕ k )/ϕ k +1 = 0 for every f: ℝ → ℝ definable in ℜ (Such sequences always exist  ...  Then (ℜ, (S)) is d-minimal. where S ranges over all subsets of cartesian powers of the range of ϕ.  ...  Proofs of Theorems 2 and 3 The reader is now assumed to be familiar with basic properties of o-minimal structures, e.g., [3, Chapters 1 and 3].  ... 
doi:10.2178/jsl/1120224720 fatcat:whndm5mvhrai7hutqdpidha4tm

Definable sets in ordered structures. III

Anand Pillay, Charles Steinhorn
1988 Transactions of the American Mathematical Society  
Thus, with the results of §1, discrete o-minimal structures lose their interest.  ...  We show that any o-minimal structure has a strongly o-minimal theory. 0. Introduction. In this paper we prove that an arbitrary o-minimal structure M is strongly o-minimal.  ...  Mixed o-minimal structures. Here we will show that an arbitrary ominimal structure M is strongly o-minimal.  ... 
doi:10.1090/s0002-9947-1988-0943306-9 fatcat:wuswpwwrvbfyxgwgbbqlrukb5q

Definable Sets in Ordered Structures. III

Anand Pillay, Charles Steinhorn
1988 Transactions of the American Mathematical Society  
Thus, with the results of §1, discrete o-minimal structures lose their interest.  ...  We show that any o-minimal structure has a strongly o-minimal theory. 0. Introduction. In this paper we prove that an arbitrary o-minimal structure M is strongly o-minimal.  ...  Mixed o-minimal structures. Here we will show that an arbitrary ominimal structure M is strongly o-minimal.  ... 
doi:10.2307/2000920 fatcat:sdzjx6j6ubc4fbyh3fjbal76ga

On the expressiveness and decidability of o-minimal hybrid systems

Thomas Brihaye, Christian Michaux
2005 Journal of Complexity  
We show that this can be done for a quite general class of hybrid systems defined on o-minimal structures.  ...  In the case of o-minimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics  ...  There exists many more interesting o-minimal structures. The main result we use on o-minimal structures is (see [vdD98, Corollary 3 .6, p. 60]): Theorem 2.13 (Uniform Finiteness).  ... 
doi:10.1016/j.jco.2004.09.003 fatcat:24vweszpuzgrbj5sd3g5iei6ia

SHAPE DESIGN OF MEMBRANE STRUCTURE USING GEOMETRIC INVARIANTS OF DISCRETE SURFACE

Kentaro HAYAKAWA, Makoto OHSAKI
2021 Journal of Structural and Construction Engineering (Transactions of AIJ)  
The surface is discretized into triangular faces.  ...  Optimization procedures are proposed for design of frame-supported membrane structures. A developable, a minimal, and their intermediate surfaces are chosen as the target surfaces.  ...  Suris: Discrete Differential Geometry: Integrable Structure, American Mathematical Society, 2008. 16) J. M. Sullivan: Curvatures of smooth and discrete surfaces, Discrete Differential Geometry.  ... 
doi:10.3130/aijs.86.772 fatcat:nspiaepgqnf27cwh6lbnywjcj4

Locally o-minimal open core [article]

Masato Fujita
2021 arXiv   pre-print
We demonstrate that the open core of a definably complete expansion of a densely linearly ordered abelian group is locally o-minimal if and only if any definable closed subset of R is either discrete or  ...  Here, the notation R denotes the universe of the original structure.  ...  Definably complete uniformly locally o-minimal structures of the second kind satisfy the additional condition which general locally o-minimal structures do not satisfy.  ... 
arXiv:2111.08197v1 fatcat:tumlgb3x4nfqvc6fkdwoqebduq

Dp-minimal expansions of discrete ordered abelian groups [article]

Erik Walsberg
2020 arXiv   pre-print
If Z is a dp-minimal expansion of a discrete ordered abelian group (Z,<,+) and Z does not admit a nontrivial definable convex subgroup then Z is interdefinable with (Z,<,+) and (Z,<,+) is elementarily  ...  Many structures whose definable sets are "tame topological" objects are dp-minimal. Dp-minimal theories include o-minimal, Cminimal, P-minimal, strongly minimal theories, and Presburger arithmetic.  ...  Throughout this paper (Z, <, +) is a discrete ordered abelian group, 1 Z is the minimal positive element of Z, O ∶= {k1 Z ∶ k ∈ Z}, and Z is an expansion of (Z, <, +).  ... 
arXiv:1903.06222v2 fatcat:d53yih5fpndwhjnmk6pkr4n5wq
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